 # Chapter 9: Simple Regression Continued

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Chapter 9: Simple Regression Continued
Hypothesis Testing and Confidence Intervals

Inferences on regression coefficients
To place confidence intervals or test hypotheses on a and b we need to know sa2, and sb2 which will be estimated by sa2 and sb2. Where s2 is estimated by:

Inferences on regression coefficients.
If the model is correct, then b/sb and a/sa are distributed as a t distribution with n-2 degrees of freedom. Confidence limits on a are: Confidence limits on b are:

Hypothesis Testing: Test on a
Ho: a = ao Ha: a ≠ ao Test Statistic Reject Ho if

Hypothesis testing: Test on b
Ho: b = bo Ha: b ≠ bo Test Statistic Reject Ho if

Hypothesis Testing: Significance of regression equation
Ho: b = 0 (equivalent to Ho: r = 0) Ha: b ≠ 0 Test statistic and rejection region same as previous test on b. If this hypothesis is not rejected then may be estimated by If r = 0 then s2 ≈ sy2 or the regression line does not explain a significant amount of the variation in Y.

Confidence Intervals on the Regression Line
Determined by first calculating the variance of ,the predicted mean of for a given Xk. The standard error of can be estimated by calculated as:

Confidence Intervals on the regression line
The variance of depends on the value of X at which the variance is being determined. Var ( ) is a minimum where Xk = and increases as Xk deviates from . Confidence limits on the regression line are:

Confidence Intervals on the regression line
Since increases as Xk - increases, the confidence intervals on Xk = are at their narrowest and widen as Xk deviates from . Confidence limits on an individual predicted value of Y would be wider than the confidence interval on the regression line, since for an individual Y, the Var (e) or s2 would have to be added to the Var( ). Thus the variance of an individual predicted value of Y would be Var( +s2).

Confidence intervals on an individual predicted value of Y
Can be calculated by the previous confidence interval equations where would be substituted for

Confidence Intervals on the standard error
Can be made by noting that (n-2)s2/s2 is distributed as a chi-squared distribution with n-2 degrees of freedom. Limits are given by: where